- The Pythagoreans considered all mathematical science to be divided into four parts:
- one half they marked off as concerned with quantity,
- the other half with magnitude;
- and each of these they posited as twofold.
- A quantity can be considered in regard to its character by itself or in its relation to another quantity,
- magnitudes as either stationary or in motion.
- Arithmetic, then, studies quantities as such,
- music the relations between quantities,
- geometry magnitude at rest,
- spherics [astronomy] magnitude inherently moving.
- from Proclus
I have to admit I failed geometry in school all because of one fatal phrase. Our problems began ‘Given that’ followed by a list of assumptions. Who, I needed to know, gave what to whom? Where did these ‘givens’ come from? I was completely stuck and my teacher was most unsympathetic. Not knowing who gave what to whom did not stop her from knowing where I should go, and that was out of her classroom. Eventually I realized that the lack of sympathy was rooted in a lack of knowledge, although I found the idea that the teachers did not know the source and foundations of what they were teaching even more disturbing than my own ignorance. Eventually I discovered that in medieval times mystery schools taught the trivium of language which were grammar, rhetoric and dialectic, and the quadrivium of numbers which were arithmetic, geometry, music, and astronomy. However I was also a failure at the mystery schools. Leaping from mystical oneness to the complexity of the Gothic cathedrals without any comprehensible steps between did not instill me with confidence. It seemed to me the focus was so much on being mysterious that the secret itself is forgotten.
Galloping off into complex geometric forms, leaving me still worried about where any of it started felt terribly inconsiderate, so I decided to start my quest into the roots of the quadrivium by looking at our assumptions about the point. The point is promoted by nearly everyone as the source of everything that follows, except for Euclid who defined the point as the place where two lines intersect (but where, oh where, do the lines come from?!). The point has no dimension, not height or length or width, or duration. On a piece of paper it appears as a dot, in the world of time and space it would be something like a black hole.
Next, I tried the circle. The circle and the dot seem interchangeable in most sources on sacred geometry. When the circle is explained, it is said to derive from the line. That seems like circular reasoning to me! The line is a terribly complicated form and it is contradictory to claim that unity arises out of such complexity. Besides, according to sacred geometry, the Vesica Pisces, or two interlocking circles, represents the universal wholeness of the dot manifesting into the world of form, the basic geometric form of the Mother Goddess giving birth to all creation. Again, there is a whole world of theology assumed in that view leaving me to ask the obvious question again. How did we get from one all-encompassing yet dimensionless and indefinable dot to two overlapping circles? How did we even get one circle at all, let alone two circles in a very defined specific relationship to one another?
Lost, I went to look up Chaos in the dictionary. Chaos, according to Webster’s, is the disorder of formless matter in infinite space. If the point exists outside of time and space, the infinite space of chaos must be the first phase in the manifestation into form. I found support for this view in the I Ching. In this view the yin, or feminine essence, is a point when at rest. When active it opens into a sphere, creating the space necessary for the creation of form. Interestingly, the pattern that results from the vibrations of the Sanskrit Mantra ‘Aum’ is rings of concentric circles.
Having gotten some space, however chaotic and undefined, my next question is ‘how do we bring order out of chaos?’ This is the fundamental question of all human mythologies, including our western sciences. Quantum physics has changed our view of the atom from something like a small solar system with a solid positively charged nucleus in the center with small planet-like electrons orbiting around it. Instead, there is a cloud of positively charged potential in the center surrounded by a cloud of negatively charged potential all around it. This is the first sphere, the first orderly relationship that appears in space. We do not have any lines, because there are not any things yet. Our sphere is still unmanifest potential. This is the pregnant possibility within the oneness.
Next, my question was how do we get a second sphere? There may be universes where infinite disorder becomes infinite order in one fell swoop, but judging by the state of my room as I write this, ours is not one of them. Making an intuitive leap and assuming that disorder is in the process of becoming order, we can say that order appears in a finite space. This leaves a whole lot of infinity still disordered. In infinite space the chances of two spheres of order bumping into one another are infinitesimally small with the most likely result should they manage to cross paths being a return to disorder.
The simplest way of creating more spheres of potentiality that are compatible with one another is to assume that orderliness is a generative process with order begetting order, creating mirror images of itself. In this scenario, one sphere creates the next in its own image, like begets like. Now, are two spheres of orderly potentiality going to wander off into infinite disorder or are they going to hang around each other interacting in an orderly fashion? I settled for the point being the moment and area that two spheres of potentiality first make contact with each other. Then I have a point that is both indefinable and filled with potential. The ancient Egyptians put the possibilities of the point into a beautifully simple poem attributed to Thoth.
- I am One that transforms into Two
- I am Two that transforms into Four
- I am Four, that transforms into Eight
- After all of this, I am One
(I apologize for no illustrations, but getting microsoft word graphics into the blog is a challenge I haven’t yet resolved.)
I am One that transforms into Two
Starting with the first sphere, and doubling it, our two spheres of potential interact showing us the Vesica Pisces (click here).
I am Two that transforms into Four
Then the whole Vesica Pisces doubles, not just a single sphere. While there are still an infinite number of potential positions in an infinite number of directions, there are now two criteria to define what we are looking for. We want first, orderly relations, second, that repeat themselves. We find ourselves with a pattern that extends itself, perhaps infinitely. (orderly relations that infinitely repeat themselves are now called fractals: click here).
I am Four, that transforms into Eight
Next, double the already doubled pair of Vesica Pisces. There are still and yet an infinite number of potential positions in an infinite number of directions, and we still have our two criteria of orderly repeating relations to define what we are looking for. Out of all the possibilities of doubling, the first repeating angle that appears is the quarter angle, and it promptly mirrors itself forming a flower with four petals. This shape is special, rarely occurring in nature. Most flowers have odd numbers of petals. We tend to know it best as the four-leafed clover, bringing us luck. Why is it lucky? Calling something lucky is often a way of remembering those things once sacred to our ancestors. This shape represents the Pythagorean’s quadrivium, the four directions essential to creation, and in the macrocosm, these are the four points of the Solar Wheel, the equinoxes, and the solstices.
After all of this, I am One
With a defined center and circumference to our sphere of potentialities, the first Vesica Pisces is now the radius defining the distance of the circumference from the center of the circle, and there we finally have a point, a true circle, and the first line. Once there is a line, since we are working in multiples of two’s, both the square and the cube are implied. This circle is the circle of life; all the action within duality happens within this arena.
Whew! No wonder people like to skip this phase and go straight to the dot and line!